Linear Algebra with Applications, 10th Edition

6.1

1. 1. 

   1. (a) ${\lambda }_1=5$, the eigenspace is spanned by ${(1,1)}^T$, ${\lambda }_2=-1$, the eigenspace is spanned by $(1,-2{)}^T$;

   2. (b) ${\lambda }_1=3$, the eigenspace is spanned by ${(4,3)}^T$, ${\lambda }_2=2$, the eigenspace is spanned by ${(1,1)}^T$;

   3. (c) ${\lambda }_1={\lambda }_2=2$, the eigenspace is spanned by ${(1,1)}^T$,

   4. (d) ${\lambda }_1=3+4i$, the eigenspace is spanned by ${(2i,1)}^T,{\lambda }_2=3-4i$, the eigenspace is spanned by ${(-2i,1)}^T$;

   5. (e) ${\lambda }_1=2+i$, the eigenspace is spanned by ${(1,1+i)}^T,{\lambda }_2=2-i$, the eigenspace is spanned by ${(1,1-i)}^T$;

   6. (f) ${\lambda }_1={\lambda }_2={\lambda }_3=0$, the eigenspace is spanned ${(1,0,0)}^T$;

   7. (g) ${\lambda }_1=2$, the eigenspace is spanned by ${(1,1,0)}^T$, ${\lambda }_2=1$, the eigenspace is spanned by ${(1,0,0)}^T,{(0,1,-1)}^T$;

   8. (h) ${\lambda }_1=1$, the eigenspace is spanned by ${(1,0,0)}^T$, ${\lambda }_2=4$, the eigenspace is spanned by ${(1,1,1)}^T$, ${\lambda }_3=-2$, the eigenspace is spanned by ${(-1,-1,5)}^T$;

   9. (i) ${\lambda }_1=2$, the eigenspace ...